Integrals of Trig Functions

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Integrals of Trig Functions

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Product of Sines and Cosines (2)
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Product of Sines and Cosines (Part 2)

If $n$ and $m$ are both even, then the tricks of the previous slide do not help. Instead, we can either integrate by parts (using the "go in a circle" trick) or use double-angle formulas. Most people find the double-angle formulas to be easier, and that's what this slide's video is about.

Since

$\cos(2x)=\cos^2(x)-\sin^2(x)$

$\qquad= 1-2\sin^2(x)$

$\qquad= 2\cos^2(x)-1,$ we can rewrite products of

$\sin^2(x)$ and

$\cos^2(x)$ in terms of

$\cos(2x)$ :

$\sin^2(x) = \frac{1-\cos(2x)}{2}$

$\cos^2(x) = \frac{1+\cos(2x)}{2}.$ This converts

$\sin^n(x)\cos^m(x)$ , with

$n$ and

$m$ even, into a polynomial in

$\cos(2x)$ of degree

$(n+m)/2$ . The odd powers of

$\cos(2x)$ can be handled as in the previous slide. To integrate the even powers, apply the double-angle trick again, getting a polynomial in

$\cos(4x)$ . Repeat as many times as necessary.